Although the time-reversal symmetry of the system is broken by tuning the coupling phase \(\theta \) , the nonreciprocity is also dependent on \(\phi \)  in the case of \(\theta \ne m\pi \) . As shown in Figs. REF (a)-(c), one can observe nearly inverse nonreciprocal transfer for \(\phi =20\pi \)  and \(\phi =21\pi \)  (\(\eta =0.56\)  and \(\eta =0.588\) ), while the transfer becomes reciprocal for \(\phi =20.5\pi \)  (\(\eta =0.574\) ) even if the coupling phase is nontrivial. This is because the coupling phases are effectively shifted from \(\pm \theta \)  to \(\pm \theta -\phi \)  by removing \(\phi \)  in the indirect coupling terms, implying that the moduli of the overall couplings of opposite directions are effectively swapped in the case of \(\phi =(2m+1)\pi \) . Note that the nonreciprocal behaviors are not exactly inverse for \(\phi =20\pi \)  and \(21\pi \)  due to different time delays before which the emitters decay exponentially [1]}.
